\(\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx\) [474]

Optimal result

Integrand size = 23, antiderivative size = 42 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{3 a \sqrt {c+a^2 c x^2}} \]

[Out]

2/3*arcsinh(a*x)^(3/2)*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783} \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{3 a \sqrt {a^2 c x^2+c}} \]

[In]

Int[Sqrt[ArcSinh[a*x]]/Sqrt[c + a^2*c*x^2],x]

[Out]

(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[c + a^2*c*x^2])

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{3 a \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{3 a \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

Integrate[Sqrt[ArcSinh[a*x]]/Sqrt[c + a^2*c*x^2],x]

[Out]

(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[c*(1 + a^2*x^2)])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {a^{2} x^{2}+1}}{3 a \sqrt {c \left (a^{2} x^{2}+1\right )}}\)	\(36\)

[In]

int(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*arcsinh(a*x)^(3/2)/a/(c*(a^2*x^2+1))^(1/2)*(a^2*x^2+1)^(1/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\sqrt {\operatorname {asinh}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(asinh(a*x)**(1/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(asinh(a*x))/sqrt(c*(a**2*x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arcsinh(a*x))/sqrt(a^2*c*x^2 + c), x)

Giac [F]

\[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\sqrt {\operatorname {arsinh}\left (a x\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]

[In]

integrate(arcsinh(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(arcsinh(a*x))/sqrt(a^2*c*x^2 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\sqrt {\mathrm {asinh}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

[In]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(1/2),x)

[Out]

int(asinh(a*x)^(1/2)/(c + a^2*c*x^2)^(1/2), x)